Manipulation of digital information organized in multidimensional finite (discrete) datasets has widespread practical applications, such as in the processing of digital two-dimensional and three-dimensional images, videostream manipulations, computer vision, data compression, and the like. Much of this dataset processing relies on mathematical transformations to achieve a desired practical change in the functional aspects of the dataset, such as the resolution of a digital image, image rotation and/or warping, the format of the image representation and storage. An example of such a transformation is the Discrete Cosine Transform (DCT), introduced in 1974 and classified into 4 basic types in 1984, which has proven to be very efficient for a large number of data processing applications. Currently the most popular version of DCT is the DCT-II implemented in the data compression standard JPEG. Earlier some versions of DCT have been also proposed for discrete interpolation, such as the modified versions of DCT-I and DCT-II. Although so far these interpolation methods have not been as popular as linear and higher-order interpolation algorithms, we believe that interpolations based on trigonometric polynomials (including cosine and sine transforms) will gain significant practical ground in the near future. This relates to interpolations by both the DCT and the continuous extensions of DCT, or CEDCT. The CEDCT has been recently derived (2004) as the one-dimensional implementation of a broad class of multidimensional discrete transforms which are based on orbit functions of compact semisimple Lie groups of various symmetries.
An important feature of the DCT-I is that it is well suited (better than other DCT and Discrete Sine Transform (DST) types) for discrete data interpolation. It can be naturally extended to form a continuous trigonometric polynomial called CEDCT. For a (N+1)-pixel data set {ƒn}n=0N which results from sampling of a function f(x) on the equidistant N-interval grid {xn}n=0N, the CEDCT function FN(x) returns the values of f(x) at the grid points x=xn. Between the grid points F(x) provides an approximation to f(x) that has a number of useful analytic properties, including convergence to the continuous sampling function f(x), localization, and differentiability, i.e., F′N→ƒ′(x) if N→∞. Because of these properties, the CEDCT provides image interpolation quality that is quite comparable with, and could even exceed, the quality of bicubic and other high-order non-adaptive interpolation algorithms. However, as with earlier data processing transformations, the CEDCT has limits in its efficiency if the zooming is done to a non-integer zooming factor m. For an integer zooming factor the CEDCT is reduced to known method of interpolation by the DCT-1 with zero-padding, the computational efficiency of which was possible to enhance using a standard fast cosine transform (FCT) algorithm.